General Ring Theory/Ring extensions

Proposition (commutative ring extension is a module):

Let be a commutative ring and let be a commutative extension ring of . Then with its own addition and the restriction of the multiplication of to is a module over .

Proof: From the axioms holding for rings, we deduce the module axioms as follows as follows: Let and . Then

  1. (distributivity)
  2. (distributivity)
  3. (commutativity of multiplication)
  4. (unit),

the ring axiom that's being used being indicated in the brackets.

Proposition (Let be a ring extension. Then the function

defines a function from ideals of to ideals of .):


Proof: Indeed, because it is certainly closed under addition and multiplication by elements of .

Proposition (Let be a ring extension, and suppose that is a multiplicative set. Then is a multiplicative set of .):


Proof: is closed under multiplication because both and are.